Buy a quality bolt action before you do anything else
So you bought a Ford Pinto, and now you wish to get a Chevy pickup? Not so fast, give the bolt actions another chance. I like vintage guns, and some good vintage options for you would be: Remington 721, Sako Finnbear, Winchester 70 pre 64, etc. For your purposes stated, I would want a bolt action for reliability and accuracy. Not that a lever or semi could not duplicate it, but not as easily, so I would rather have a bolt action myself.
If you didn't want a vintage Remington, Sako or Winchester, you could look for a Belgium made FN Browning Hi Power rifle
This one is a safari grade. It wears a Weaver V9. It is not a salt wood gun. The quality is excellent. It'll cost more coin than a Remington 770, but its also in a different league. Beautiful wood, blue, a slick action, excellent trigger - the quality just radiates from it when you hold it in your hands.
Remember, for the most part, a cheaper gun often costs you more money in the end. Between upgrades, repairs, and the eventual sale (presumably at a loss) to get something better, you are better off to buy quality in the beginning. I have lived by that maxim, and never looked back. Occasionally you can find a gun that has both quality and value, but not very often.